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Definitions:

A progression of numbers is called a sequence. E.g., 2, 4, 8, 16,…, 1024 The
sequence can be finite if it ends or infinite if it is unending. In our example,
the sequence ends with the 10th term, 1024, and is therefore a finite sequence.

A term of a sequence (or series) is an individual number in the progression.
In our example above, the 3rd term is 8.

A Series is the sum of a progression of numbers. E.g., 2+4+8+…+1024. Often
we are interested in the particular number that this sum equals.

Ratios in a sequence are the numbers obtained by dividing a term by its
previous term. In our example, the ratios are:
 4/2=2
 8/4=2
 etc.

An Geometric Sequence is a sequence in which all the ratios are the same.
Our example 2, 4, 8, …, 1024 is a geometic sequence because the ratios all
equal 2. A second example, 1, 3, 5, 7, …, 45 is not a geometic sequence
because its ratios aren’t all the same.

The General Term of a sequence is a kind of basket that represents all of the
terms of the sequence at once. In our example 2, 4, 8, … the general term is
{2x }. When x=3, we see that 23 = 8 is the third term. The general term makes
it easy to predict that the 10th term is 210 = 1024 or that the 1034th term is
21034.
Technique for Geometic Sequences
1. Find the ratio. This identifies the power (exponential) function that your
sequence is a variation on, and hence tells you what base to raise to the x
power. In the example 3, 6, 12, 24, … this ratio would be 2 telling us that this
sequence is a variation on the 2x function: 2, 4, 8, 16,…
2. Find what term zero of the sequence would be. In our example above, term
zero is what would come before the 3 in 3, 6, 12, 24,…. This would be 1.5.
This is what must be multiplies by the power function of x found in step 1.
3. Put the above two parts together and you get the general term: {1.5(2x)}.
4. Now you are in a position to use the general term to make predictions. For
instance, the 10th term is 1.5(210) = 1.5(1024) = 1536. We can also find out
what term number is 192 by solving the equation 1.5(2x)=192 to find that
2x=128 and hence x=7.
Find the 10th term of the sequence
20, 40, 80, …
10,
Ratio:
2
Term 0:
5
General Term: 5(2x)
Term 10:
5(210)=5(1024)=5120
Find the 20th term of the sequence
18, 54, 162, …
Ratio:
3
Term 0:
2
General Term: 2(3x)
6,
Term 20:
2(320)=2(3486784401)=6973568802
Find the 6th term of the sequence
200, 2000, …
20,
Ratio:
Term 0:
General Term:
Term 6:
Find the 10th term of the sequence
9, 27, 81, …
Ratio:
Term 0:
General Term:
Term 10:
3,
Find the 11th term of the sequence
24, 12, 6, …
48,
Ratio:
Term 0:
General Term:
Term 11:
Find the 9th term of the sequence
576, 192, 64, …
1728,
Ratio:
Term 0:
General Term:
Term 9:
Find the 10th term of the sequence
5, 25, 125, …
Ratio:
Term 0:
General Term:
Term 10:
1,
Find the 8th term of the sequence
54, 36, 24, …
Find the 13th term of the sequence
2/3, 4/3, 8/3, …
81,
1/3,
Find the 10th term of the sequence
2/3, 4/9, 8/27, …
1,
Find the 10th term of the sequence
1/2, 2, 8, …
1/8,
Answers
Page
No.
Example
No.
Ratio
Term 0
General Term
nth Term
3
1
2
5
5(2x)
5120
2
3
2
2(3x)
6973568802
1
10
2
2(10x)
2000000
4
5
6
2
3
1
3x
59049
3
1/2
96
96((1/2)x)
3/64
1
1/3
5184
5184((1/3) x)
64/243
2
5
1/5
(1/5)(5x)
1953125
3
2/3
1
2
1/6
(1/6)(2x)
4096/3
2
2/3
3/2
(2/3)((3/2)x)
512/19683
3
4
1/32
(1/32)(4x)
32768
243/2 (243/2)((2/3)x)
64/243
Green Valley School
389 Pembroke Street
Pembroke, NH 03275
(603) 485-8550